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An Overview of Sieve Methods Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 2, 67 - 80 An Overview of Sieve Methods R. A. Mollin Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada, T2N 1N4 URL: http://www.math.ucalgary.ca/˜ ramollin/ [email protected] Abstract We provide an overview of the power of Sieve methods in number theory meant for the non-specialist. Mathematics Subject Classification: Primary: 11N35; Secondary: 11- 02; 11N36 Keywords: Sieves, open problems 1 Sieves Some of the following is adapted from [11]. Sieve methods are used in fac- toring, recognizing primes, finding natural numbers in arithmetic progression whose common difference are primes, or generally to estimate the cardinali- ties of various sets defined by the use of multiplicative properties. Recall that use of a sieve or sieving is a process whereby we find numbers via searching up to a prescribed bound and eliminate candidates as we proceed until only the desired solution set remains. In other words, sieve theory is designed to estimate the size of sifted sets of integers. For instance, sieves may be used to attack the following open problems, for which sieve methods have provided some advances. (a) (The Twin Prime Conjecture) There are infinitely many primes p such that p + 2 is also prime. (b) (The Goldbach Conjecture) Every even integer n>2 is a sum of two primes. 68 R. A. Mollin (c) (The p = n2 + 1 Conjecture) There are infinitely many primes p of the form p = n2 +1. (d) (The q = 4p + 1 Conjecture) There are infinitely many primes p such that q =4p + 1 is also prime. (e) (Artin’s Conjecture) For any nonsquare integer a ∈{−1, 0, 1}, there exist infinitely many primes p such that a is a primitive root modulo p. Indeed, in 1986, Heath-Brown [8] used sieving methods to advance the Artin conjecture to within a hair of a solution when he proved that with the possible exception of at most two primes, there are infinitely many primes q such that p is a primitive root modulo q. Thus, sieve methods are important to review for their practical use in number theory and the potential for solutions of outstanding problems such as the above. The fundamental goal of sieve theory is to produce upper and lower bounds for cardinalities of sets of the type, S(S, P,y)={n ∈ S : p n implies p>yfor all p ∈ P}, (1) where S is a finite subset of N, P is a subset of P, the set of all primes, and y is a positive real number. Example 1 Let √ S = {n ∈ N : n ≤ x} and x<y≤ x. Then |S(S, P,y)| = {n ≤ x : p n implies p>y} = π(x) − π(y)+1, one more than the number of primes between x and y. To illustrate (1) more generally, we begin with what has been called “the oldest nontrivial algorithm that has survived to the present day.” From antiq- uity, we have the Sieve of Eratosthenes, which is covered in a first course in number theory—see [10, Example 1.16, p. 31], which sieves to produce primes to a chosen bound. However, as discussed therein, this sieve is highly ineffi- cient. Indeed, since in order to determine the primes up to some bound using ∈ N √this sieve for n , one must check for divisibility by all primes not exceeding n, then the sieve of Eratosthenes has complexity O(n loge n)(loge loge n)), An overview of Sieve methods 69 which even using the world’s fastest computers, this is beyond hope for large integers as a method for recognizing primes. Yet there is a formulation of this sieve that fits nicely into the use of arithmetic functions, and has appli- cations as a tool for modern sieves, so we present that here for completeness and interests sake. Recall that the M¨obius function μ(d) is defined by 1ifn =1, μ(n)= 0ifn is not squarefree, k k (−1) if n = j=1 pj where the pj are distinct primes. Also, let ω(d) denote the number of distinct prime divisors of d, and P the set of all primes. Theorem 1 Eratosthenes’ Sieve Let P = {p1,p2,... ,pn}⊆P be a set of distinct primes and let S ⊆ N with |S| < ∞. Denote by S the number of elements of S not divisible by any of the pj’s and by Sd the number of elements of S divisible by a given d ∈ N. Then S = μ(d)Sd. d|p1p2···pn Moreover, For m =1, 2,... ,n/2 , we have μ(d)Sd ≤ S ≤ μ(d)Sd, d|p p ···p d|p p ···p 1 2 n 1 2 n ω(d)≤2m−1 ω(d)≤2m where (1) is called Eratosthenes’ sieve. Proof. See [12, Corollary 2, p. 147]. 2 For instance, an application of Theorem 1 is that it may be used to prove the following result on the number of primes less than a certain bound, first proved in 1919, by the Norwegian mathematician Viggo Brun (1882–1978). Theorem 2 Brun’s Theorem If n ∈ N and B2n(x) denotes the number of primes p ≤ x for which |p +2n| is also prime, then 2 −2 B2n(x)=O(x(loge loge x) loge x). 70 R. A. Mollin Proof. See [12, Theorem 4.3, p. 148]. 2 Theorem 2 has as a special case, implications for the twin prime conjecture as follows. Corollary 1 Brun’s Constant Let Q be the set of all primes p such that p +2 is also prime, then 2 x(loge loge x) 1 << 2 , loge x p∈É p≤x and the series 1 = B (2) p p∈É is convergent, where (2) is called Brun’s constant. Proof. See [12, Corollary, p. 152]. 2 Remark 1 We do not know if Q in Corollary 1 is finite or not since its infini- tude would be the twin prime conjecture. We do know that the sum of the reciprocals of all primes diverges, but since the series (2) converges, this is not a proof of the conjecture since we would need divergence to get the infinitude. The behaviour of the two series does tell us that, although the twin prime conjecture may be true, the twin primes must be appreciably less dense than the entire set of primes. Brun’s result, that the reciprocals of twin primes converges, is one of the centerpiece achievements of sieve theory. The value of Brun’s constant is B ≈ 1.9021605824, with an error within ±0.000000003, computed by Thomas R. Nicely in 1999. It is worth noting the now famous fact that, in 1995, Nicely was doing computa- tions on Brun’s constant which led him to discover a flaw in the floating-point arithmetic of the Pentium computer chip, costing literally millions of dollars to its manufacturer Intel—see http://www.trnicely.net/twins/twins2.html. Theorem 1 tells us that the sieve of Eratosthenes investigates the function S(S, P,x)= 1, where Π = p n∈Ë p∈È gcd(n,Π)=1 p≤x An overview of Sieve methods 71 via the equality S(S, P,x)= μ(d)= μ(d)Sd. d|n n∈Ë d|Π d|Π The general basic sieve problem emanates from this, namely find arithmetic functions λ(d):N → R and λu(d):N → R with 1 if gcd(n, Π) = 1, λ(d) ≤ 0 if gcd(n, Π) > 1, d|n d|Π and 1 if gcd(n, Π) = 1, λu(d) ≥ 0 if gcd(n, Π) > 1, d|n d|Π such that λ(d)Sd = λ(d) ≤ S(S, P,x) ≤ λu(d)= λu(d)Sd. (3) d|n d|n n∈Ë d|Π n∈Ë d|Π d|Π d|Π Now we interpret the above in terms of what Selberg did to create his fa- mous sieve and how Theorem 1 comes into play. With the notation of Theorem 1 still in force, we add that P denotes the product of the primes in P, |S| = N, and call the following Selberg’s condition on S. There exists a multiplicative function f(d) such that if d P , then f(d) Sd = N + R(d), (4) d where |R(d)|≤f(d) and d>f(d) > 1. With the Selberg condition plugged into the right-hand side of (3), we have λu(d)f(d)N |S(S, P,x)|≤ + λu(d)R(d) d|Π d d|Π ⎛ ⎞ λu(d)f(d) ⎝ ⎠ = N + O |λu(d)R(d)| . (5) d|Π d d|Π Selberg’s sieve arose from his attempts to minimize (5) subject to Selberg’s condition (4). Theorem 1 comes into play again in that it is used in the proof of the following, first proved by Selberg [13] in 1947. The following is considered to be the fundamental theorem concerning Selberg’s sieve, which for the above-cited reasons, is often called Selberg’s upper bound sieve. 72 R. A. Mollin Theorem 3 Selberg’s Sieve Let P be a finite set of primes, P denoting their product, S ⊆ N with |S| = N ∈ N, such that S satisfies Selberg’s condition (4), and let S = S(S, P,x) be the number of elements of S not divisible by primes p ∈ P with p ≤ x where x>2. If for p P , we have that f(p) > 1, μ(n/d)d g(n)= , d|n f(d) and −1 Qx = g (d), d|P d≤x then −2 N f(p) S ≤ + x2 1 − .
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